TAILIEUCHUNG - Appendix 5 Spontaneous Emission Term and Factors

Consider the spontaneous emission term describing the contribution of the spontaneous emission to the laser oscillation, which appears as the final term on the right-hand side of the rate equation for the photon density (Eq. ()). Since this term represents the component of the spontaneous emission belonging to the same mode as the laser oscillation (angular frequency !m), it is closely related to the gain for the oscillation mode. Consider a laser of index-guiding type, and let E ¼ E(r) ¼ E(x, y)E(z) be the complex electric field of the oscillation mode. . | Appendix 5 Spontaneous Emission Term and Factors Consider the spontaneous emission term describing the contribution of the spontaneous emission to the laser oscillation which appears as the final term on the right-hand side of the rate equation for the photon density Eq. . Since this term represents the component of the spontaneous emission belonging to the same mode as the laser oscillation angular frequency a m it is closely related to the gain for the oscillation mode. Consider a laser of index-guiding type and let E E r E x y E z be the complex electric field of the oscillation mode. The field is normalized in the similar manner as Eq. so that the optical energy in the resonator corresponds to the energy of a photon f 0 r g E r- 2dv m Jc 2 Then from Eqs and the transition probability relevant to the photons of this mode can be written as wabs wstm n n wspt p . .9 E r 2 RCE fi m - EJ where n is the number of the photons in the resonator. Using the direct-transition model the net number of the stimulated emission transition per unit volume of the active region per unit time can be calculated by integrating wstm multiplied by 1 2p3 f2 f1 dk. The average value of E r 2 in the active region is given by p G 2fi m M ng Va Copyright 2004 Marcel Dekker Inc. r Ra E r 2dV Rc E r 2dV where Va is the volume of the active region r is the confinement factor and use has been made of Eq. . Replacing E in Eq. by the average given by Eq. and calculating the relative time variation in the mode photon number in the resonator of volume Va we obtain an expression for the mode gain 2 pe n . . . . GG m r-------- M 2 f2 - 1 Pr h m nrngsom2rnm In the derivation of the above expression use has been made of Eqs . This result is consistent with that obtained by rewriting the material gain g given by Eq. in G vgg and then in the mode gain TG. Let Rsp a m be the number of photons spontaneously emitted per unit time in

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